Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Iterated Deriv sub

W1.iteratedDeriv_sub

analytic number theory prime numbers asymptotics sobolev

Plain-language statement

For two $n$ -times continuously differentiable functions $f$ and $g$ , the $n$ th iterated derivative distributes over subtraction: $(f-g)^{(n)}=f^{(n)}-g^{(n)}.$

Exact Lean statement

lemma iteratedDeriv_sub {f g : ℝ → E} (hf : ContDiff ℝ n f) (hg : ContDiff ℝ n g) :
    iteratedDeriv n (f - g) = iteratedDeriv n f - iteratedDeriv n g

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma iteratedDeriv_sub {f g : ℝ → E} (hf : ContDiff ℝ n f) (hg : ContDiff ℝ n g) :    iteratedDeriv n (f - g) = iteratedDeriv n f - iteratedDeriv n g := by  induction n generalizing f g with  | zero => rfl  | succ n ih =>    have hf' : ContDiff ℝ n (deriv f) := hf.iterate_deriv' n 1    have hg' : ContDiff ℝ n (deriv g) := hg.iterate_deriv' n 1    have hfg : deriv (f - g) = deriv f - deriv g := by      ext x ; apply deriv_sub      · exact (hf.differentiable (by simp)).differentiableAt      · exact (hg.differentiable (by simp)).differentiableAt    simp_rw [iteratedDeriv_succ', ← ih hf' hg', hfg]

Project: Prime Number Theorem and More
License: Apache-2.0
Commit: a93551347dce
Source: PrimeNumberTheoremAnd/Sobolev.lean:144-155

Reuse this declaration

Bring the exact result into your workflow

The import identifies the source module. Your project still needs the pinned package dependency shown on this page.

What this badge means

This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.

Continue in this project

Related declarations

Project-declaredLean 4.32.0

Admissible bound mono

admissible_bound.mono

Plain-language statement

For positive parameters $A,B,C,R$ , the classical error-bound function $A\left(\frac{\log x}{R}\right)^B\exp\!\left(-C\sqrt{\frac{\log x}{R}}\right)$ is nonincreasing once $x\ge \exp\!\left(R(2B/C)^2\right)$ .

analytic number theoryprime numbersasymptotics

Source project: Prime Number Theorem and More

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Analytic On div Removable zero

AnalyticOn_divRemovable_zero

Plain-language statement

Let $f$ be analytic on an open set $s$ containing $0$ , and suppose $f(0)=0$ . Define $g(z)=f(z)/z$ for $z\ne0$ and $g(0)=f'(0)$ . Then the apparent singularity at $0$ is removable and $g$ is analytic throughout $s$ .

analytic number theoryprime numbersasymptotics

Source project: Prime Number Theorem and More

Person-level attribution pending.

View proof record

Project-declaredLean 4.32.0

Analytic On div Removable zero closed Ball

AnalyticOn_divRemovable_zero_closedBall

Plain-language statement

Suppose $R>0$ and $f$ is analytic on the closed disc $|z|\le R$ with $f(0)=0$ . Define $g(z)=f(z)/z$ for $z\ne0$ and $g(0)=f'(0)$ . Then $g$ is analytic on the entire closed disc, including at the removed singularity.

analytic number theoryprime numbersasymptotics

Source project: Prime Number Theorem and More

Person-level attribution pending.

View proof record